The 601 Waiver Request
Form I-601 can be used to apply for a waiver if you are ineligible to adjust your status in the U.S. or to enter the United States as an immigrant for specific grounds of inadmissibility. This may also affect your ability to enter the U.S. under specific other visa categories.
The I-601 waiver is used by applicants applying from outside the U.S. Once granted, it waives the unlawful presence grounds of inadmissibility so that foreign nationals can successfully return to the United States as lawful permanent residents.
The 601A Waiver Request
The 601A waiver request is used by applicants who entered the U.S. illegally and have green card family members or immediate U.S. citizens. Immigrant visa applicants seeking a green card are required to depart from the United States and apply at a U.S. embassy or consulate as part of the immigrant visa process.
In addition, they are typically required to attend the immigrant visa interview abroad in their home country. The problem is that if they leave the U.S. to do so, they could fall foul of the three-year or ten-year ban on trying to re-enter the U.S.
Once the 601A waiver request is granted, applicants and their immediate relatives may re-enter the U.S. on immigrant visas without fear of being banned.